Narrow Results

In this paper, we introduce the definition of pre-Gel’fand–Dorfman algebras and present several constructions. Moreover, we show that quadratic left-symmetric conformal algebras correspond to pre-Gel’fand–Dorfman algebras. Then we investigate the simplicities and central extensions of quadratic left-symmetric conformal algebras by a one-dimensional center from the point of view of pre-Gel’fand–Dorfman algebras. We show that under some conditions, central extensions of quadratic left-symmetric conformal algebras by a one-dimensional center can be characterized by four bilinear forms on pre-Gel’fand–Dorfman algebras. Several methods to construct simple quadratic left-symmetric conformal algebras from pre-Gel’fand–Dorfman algebras are also given.

This paper investigates the algebraic properties of the Bondi–Metzner–Sachs (BMS) superalgebra, as initially introduced by G. Barnich, L. Donnay, J. Matulich and R. Troncoso, Asymptotic symmetries and dynamics of 3D flat supergravity, J. High Energy Phys.2014 (2014) 071. We introduce a new realization of the BMS superalgebra using the Balinsky–Novikov construction. Utilizing this approach, we prove that the BMS superalgebra constitutes the unique minimal supersymmetric extension of the BMS algebra in three dimensions. Additionally, we compute the low-order cohomology groups of the BMS superalgebra and classify its derivations, central extensions, and automorphisms.

Fix $a,b\in ℂ$, let $LW(a,b)$ be the loop $W(a,b)$ Lie algebra over $ℂ$ with basis $\{L_{\alpha ,i},I_{\beta ,j}|\alpha ,\beta ,i,j\in \mathbb{Z}\}$ and relations $[L_{\alpha ,i},L_{\beta ,j}]=(\alpha -\beta )L_{\alpha +\beta ,i+j},[L_{\alpha ,i},I_{\beta ,j}]=-(a+b\alpha +\beta )I_{\alpha +\beta ,i+j},[I_{\alpha ,i},I_{\beta ,j}]=0$, where $\alpha ,\beta ,i,j\in \mathbb{Z}$. In this paper, a formal distribution Lie algebra of $LW(a,b)$ is constructed. Then the associated conformal algebra $CLW(a,b)$ is studied, where $CLW(a,b)$ has a $ℂ[\partial ]$-basis $\{L_i,I_j|i,j\in \mathbb{Z}\}$ with $\lambda$-brackets $[L_i{}_{\lambda }{L}_j]=(\partial +2\lambda )L_{i+j},[L_i{}_{\lambda }{I}_j]=(\partial +(1-b)\lambda )I_{i+j}$ and $[I_i{}_{\lambda }{I}_j]=0$. In particular, we determine the conformal derivations and rank one conformal modules of this conformal algebra. Finally, we study the central extensions and extensions of conformal modules.

In this paper, we study the structure theory of two classes of Lie superalgebras ${𝒮}(q)$ of Block type, where $q$ is a positive integer. In particular, the automorphism groups, derivation superalgebras and central extensions of ${𝒮}(q)$ are completely determined.

This paper deals with the classification problems of Leibniz central extensions of linear deformations of a Lie algebra. It is known that any n-dimensional filiform Lie algebra can be represented as a linear deformation of n-dimensional filiform Lie algebra μ_n given by the brackets [e_i, e₀] = e_i+1, i = 0,1,…,n - 2, in a basis {e₀, e₁,…,e_{n - 1}}. In this paper we consider a linear deformation of μ_n and its Leibniz central extensions. The resulting algebras are Leibniz algebras, this class is denoted here by Ced(μ_n). We choose an appropriate basis of Ced(μ_n) and give general isomorphism criteria. By using the isomorphism criteria, one can classify the class Ced(μ_n) for any fixed n. Two relevant maple programs are provided.

In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basis, identify the elementary base changes and describe the behavior of structure constants under these base changes, then combining them the isomorphism criterion is given. The final result of calculations for one particular case also is provided.

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

A finite group G is said to be a minimal non- group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of p-groups H of odd order with d(H) = 2 and c(H) = 2. Based on the classification of H, minimal non-p-groups G are classified for p > 3. If p > 3, then we have G₃ ≅ C_p or G₃ ≅ C_p × C_p. In this paper, we deal with the case when G₃ ≅ C_p. In another paper [A classification of finite p-groups whose proper subgroups are of class ≤ 2(II), accepted] we deal with the case when G₃ ≅ C_p × C_p.

A finite group G is said to be a minimal non- group if G is not a group of class ≤ n whose proper subgroups are of class ≤ n. In this paper, we give a complete classification of minimal non-p-groups G with G₃ ≅ C_p × C_p for p > 3.

We introduce the notion of Fitting characters for arbitrary finite groups, and prove that under some conditions the product of these characters is irreducible and the unique factorization of this form also holds. Moreover, we show that any nonlinear quasi-primitive character of solvable groups can be uniquely factored (up to multiplication by linear characters) as the product of certain Fitting characters on some extension groups.

In this paper, we give a complete algebraic classification of $5$-dimensional complex nilpotent associative commutative algebras.

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

We prove that a nonaffine latin quandle (also known as left distributive quasigroup) of order $2^k$ exists if and only if $k=6$ or $k\ge 8$. The construction is expressed in terms of central extensions of affine quandles.

We give algebraic and geometric classifications of complex four-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of three rigid algebras and two one-parametric families of algebras.

We give the classification of $5$- and $6$-dimensional complex one-generated nilpotent assosymmetric algebras.

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let $V(FG)$ denote the group of normalized units in FG. The anti-automorphism $g\mapsto g^{-1}$ of G extends linearly to an anti-automorphism $a\mapsto a^{\ast }$ of FG. An element $u\in V(FG)$ is called unitary if $u^{\ast }=u^{-1}$. All unitary elements of $V(FG)$ form a subgroup which is denoted by $V_{\ast }(FG)$. If p is odd, the order of $V_{\ast }(FG)$ is $|F{|}^{(|G|-1)/2}$. However, to compute the order of $V_{\ast }(FG)$ still is open when $p=2$. In this paper, the order of $V_{\ast }(FG)$ is computed when G is a nonabelian $2$-group given by a central extension of the following form:

We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras associated to a central extension of Leibniz n-algebras by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra.

Let be the Lie algebra over C with basis {L_m,n | m,n ∈ Z, n ≥ 0} and relations [L_m,n,L_m1,n1] =((n+1)m₁-(n₁+1)m) L_m+m1,n+n1. In this paper, we determine the second cohomology group and the second Leibniz cohomology group of .

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra , introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that is an infinite-dimensional complete Lie algebra, and the universal central extension of in the category of Leibniz algebras is the same as that in the category of Lie algebras.